I think that I should get . I see we know this is even. Then there is work to show that is times the sum of the residues of in the upper half plane. Then I found that

where Then the zeros in the upper half plane occur at .

Then I used

where Then when I add these residues up I did not get the answer above, I got

That seems to be pretty much correct, but it's not obvious how to transform your answer to the given one. I would take the residues as , where k = 1, 3, 5, 7. Their negative-fifth powers are , where k = 1, 7, –3, –5. The sum of the first two of these is , and the sum of the other two is . So the sum of the four residues is (addition formula for trig functions!). Now use and to see that times the sum of the residues is .

That is the integral from to , so you need to divide by 2 to get the integral from 0 to .